Faculty of Actuarial Science & Insurance Seminar with Anna Maria Gambaro, Universitá del Piemonte Orientale,

Time Consistent Optimal Asset Allocation for Life Insurance Funds

Abstract
In this work, we propose a dynamic consistent optimization problem for a portfolio of life insurance policies, in the Solvency II directive framework. In [Asanga et al., 2014], the authors use Solvency indicators to find the optimal asset allocation of non-life insurance funds, minimising the Solvency Capital Requirement (SCR), in a one-period model. [Christiansen and Niemeyer, 2014] propose a dynamic formulation of the SCR, using the dynamic value at risk (VaR). However, the dynamic-VaR is not a time consistent risk measure (see for instance, [Acciaio and Penner, 2010]). For non-life insurance funds and in case of a single liability cash-flow at maturity, [Devolder and Lebégue, 2017] analyse the time-consistent dynamic formulation of the SCR using the iterated-VaR and the iterated conditional tail expectation.  However, the iterated formulation of the SCR has some important drawbacks. [Devolder and Lebégue, 2017] show that using iterated risk measures, the SCR becomes quite expensive for long term liabilities and it may explode in some circumstances. Moreover, the iterated-SCR is not compliant with the Solvency II directive, in fact, it does not answer to the regulator request: how much is the capital to be held by insurance to meet his obligations over the following year?  We extend the literature in various directions. Firstly, we consider the framework of life insurance liabilities with multiple cash-flows. Secondly, starting from the static definition of the SCR in [Christiansen and Niemeyer, 2014], we propose a dynamic version of the SCR, that is time consistent, in agreement with the regulators directive and that encompass the drawbacks of the iterated formulation. Moreover, following the work of [Shapiro, 2009] on dynamic risk averse stochastic programming problems, we formulate a time consistent asset allocation problem based on the SCR minimization.

Finally, we apply the optimization problem to the case study of with-profit life insurance funds, which is analysed in [Gambaro et al., 2018] from the market-consistent valuation perspective.